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Calculus 3 : Vectors and Vector Operations

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Example Questions

Example Question #24 : Vector Addition

Find the sum of the vectors:

$\left \langle 3, 2, e^2 \right \rangle + \left \langle 0, 0, 1 \right \rangle + \left \langle 0, 0, 0 \right \rangle$ $\displaystyle \left \langle 3, 2, e^2 \right \rangle + \left \langle 0, 0, 1 \right \rangle + \left \langle 0, 0, 0 \right \rangle$

Possible Answers:

$\left \langle 3, 0, e^2+1 \right \rangle$ $\displaystyle \left \langle 3, 0, e^2+1 \right \rangle$

$\left \langle 0, 2, 0 \right \rangle$ $\displaystyle \left \langle 0, 2, 0 \right \rangle$

$\left \langle 0, 0, 0 \right \rangle$ $\displaystyle \left \langle 0, 0, 0 \right \rangle$

$\left \langle 3, 2, e^2+1 \right \rangle$ $\displaystyle \left \langle 3, 2, e^2+1 \right \rangle$

Correct answer:

$\left \langle 3, 2, e^2+1 \right \rangle$ $\displaystyle \left \langle 3, 2, e^2+1 \right \rangle$

Explanation:

To add vectors, we simply add their components. For example, $\left \langle x, y \right \rangle+\left \langle a, b \right \rangle=\left \langle x+a, y+b \right \rangle$ $\displaystyle \left \langle x, y \right \rangle+\left \langle a, b \right \rangle=\left \langle x+a, y+b \right \rangle$.

Our answer is

$\left \langle 3, 2, e^2+1 \right \rangle$ $\displaystyle \left \langle 3, 2, e^2+1 \right \rangle$

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Example Question #25 : Vector Addition

Compute the sum of the vectors $\left \langle 0,-2,1\right \rangle$ $\displaystyle \left \langle 0,-2,1\right \rangle$ and $\left \langle 0,3,-7\right \rangle$ $\displaystyle \left \langle 0,3,-7\right \rangle$.

Possible Answers:

$\left \langle 0,-1,6\right \rangle$ $\displaystyle \left \langle 0,-1,6\right \rangle$

$\left \langle -1,0,0\right \rangle$ $\displaystyle \left \langle -1,0,0\right \rangle$

$\left \langle 0,1,-6\right \rangle$ $\displaystyle \left \langle 0,1,-6\right \rangle$

$\left \langle 2,-1,0\right \rangle$ $\displaystyle \left \langle 2,-1,0\right \rangle$

Correct answer:

$\left \langle 0,1,-6\right \rangle$ $\displaystyle \left \langle 0,1,-6\right \rangle$

Explanation:

The formula for the sum of two vectors a and b is $a+b=\left \langle a_1+b_1,a_2+b_2,a_3+b_3\right \rangle$ $\displaystyle a+b=\left \langle a_1+b_1,a_2+b_2,a_3+b_3\right \rangle$. Using the numbers we have, we get the resultant as $\left \langle 0+0,-2+3,1-7\right \rangle=\left \langle 0,1,-6\right \rangle$ $\displaystyle \left \langle 0+0,-2+3,1-7\right \rangle=\left \langle 0,1,-6\right \rangle$.

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Example Question #26 : Vector Addition

Solve:

$\left \langle 2, 5\cos^2(x), 0\right \rangle + \left \langle y, 5\sin^2(x), 2\right \rangle$ $\displaystyle \left \langle 2, 5\cos^2(x), 0\right \rangle + \left \langle y, 5\sin^2(x), 2\right \rangle$

Possible Answers:

$\left \langle 2+y, 1, 2\right \rangle$ $\displaystyle \left \langle 2+y, 1, 2\right \rangle$

$\left \langle 2+y, 5, 2\right \rangle$ $\displaystyle \left \langle 2+y, 5, 2\right \rangle$

$\left \langle 2y, 25\sin^2(x)\cos^2(x), 0\right \rangle$ $\displaystyle \left \langle 2y, 25\sin^2(x)\cos^2(x), 0\right \rangle$

9+y $\displaystyle 9+y$

Correct answer:

$\left \langle 2+y, 5, 2\right \rangle$ $\displaystyle \left \langle 2+y, 5, 2\right \rangle$

Explanation:

To add vectors, we simply add the components. For example:

$\left \langle a, b, c\right \rangle+ \left \langle x, y, z\right \rangle= \left \langle ax+by+cz\right \rangle$ $\displaystyle \left \langle a, b, c\right \rangle+ \left \langle x, y, z\right \rangle= \left \langle ax+by+cz\right \rangle$

Our final answer is

$\left \langle 2+y, 5, 2\right \rangle$ $\displaystyle \left \langle 2+y, 5, 2\right \rangle$

Note that a Pythagorean identity was used when combining the "y" terms.

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Example Question #231 : Vectors And Vector Operations

Solve:

$\left \langle 5, 3x+5\right \rangle+\left \langle e^x, z\right \rangle$ $\displaystyle \left \langle 5, 3x+5\right \rangle+\left \langle e^x, z\right \rangle$

Possible Answers:

$\left \langle 5+e^x, 3x+5+z\right \rangle$ $\displaystyle \left \langle 5+e^x, 3x+5+z\right \rangle$

$\left \langle 5e^x, z(3x+5)\right \rangle$ $\displaystyle \left \langle 5e^x, z(3x+5)\right \rangle$

$\left \langle 5+e^x, 3x+z\right \rangle$ $\displaystyle \left \langle 5+e^x, 3x+z\right \rangle$

$\displaystyle 10+e^x+3x+z$

Correct answer:

$\left \langle 5+e^x, 3x+5+z\right \rangle$ $\displaystyle \left \langle 5+e^x, 3x+5+z\right \rangle$

Explanation:

To add two vectors, we simply add the corresponding components (for example, $\left \langle a, b\right \rangle+ \left \langle c, d\right \rangle= \left \langle a+c, b+d\right \rangle$ $\displaystyle \left \langle a, b\right \rangle+ \left \langle c, d\right \rangle= \left \langle a+c, b+d\right \rangle$)

Our answer is therefore

$\left \langle 5+e^x, 3x+5+z\right \rangle$ $\displaystyle \left \langle 5+e^x, 3x+5+z\right \rangle$

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Example Question #28 : Vector Addition

Find the sum of the two vectors.

u=<1,5,7> $\displaystyle u=< 1,5,7>$

v=<8,1,3> $\displaystyle v=< 8,1,3>$

Possible Answers:

<9,6,10> $\displaystyle < 9,6,10>$

<9,6,5> $\displaystyle < 9,6,5>$

<3,6,5> $\displaystyle < 3,6,5>$

<-5, 4, 4> $\displaystyle < -5, 4, 4>$

Correct answer:

<9,6,10> $\displaystyle < 9,6,10>$

Explanation:

The sum of two vectors

a=<a_1, a_2, a_3> $\displaystyle a=< a_1, a_2, a_3>$

b=<b_1, b_2, b_3> $\displaystyle b=< b_1, b_2, b_3>$

is defined as

a+b=<a_1+b_1, a_2+b_2, a_3+b_3> $\displaystyle a+b=< a_1+b_1, a_2+b_2, a_3+b_3>$

For the vectors in this problem

u+v=<1+8, 5+1, 7+3>=<9, 6, 10> $\displaystyle u+v=< 1+8, 5+1, 7+3>=< 9, 6, 10>$

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Example Question #29 : Vector Addition

Find the sum of the two vectors, u+v $\displaystyle u+v$.

u=<2,3,7> $\displaystyle u=< 2,3,7>$

v=<4,5,2> $\displaystyle v=< 4,5,2>$

Possible Answers:

<6,8,9> $\displaystyle < 6,8,9>$

<5,7,12> $\displaystyle < 5,7,12>$

<-2,-2,5> $\displaystyle < -2,-2,5>$

<4,1,3> $\displaystyle < 4,1,3>$

Correct answer:

<6,8,9> $\displaystyle < 6,8,9>$

Explanation:

The sum of two vectors

a=<a_1, a_2, a_3> $\displaystyle a=< a_1, a_2, a_3>$

b=<b_1, b_2, b_3> $\displaystyle b=< b_1, b_2, b_3>$

is defined as

a+b=<a_1+b_1, a_2+b_2, a_3+b_3> $\displaystyle a+b=< a_1+b_1, a_2+b_2, a_3+b_3>$

For the vectors in this problem

u+v=<2+4, 3+5, 7+2>=<6, 8, 9> $\displaystyle u+v=< 2+4, 3+5, 7+2>=< 6, 8, 9>$

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Example Question #30 : Vector Addition

Find the difference between $\left \langle 30,15,100\right \rangle$ $\displaystyle \left \langle 30,15,100\right \rangle$ and $\left \langle 2,25,99\right \rangle$ $\displaystyle \left \langle 2,25,99\right \rangle$.

Possible Answers:

$\left \langle 28,-10,1\right \rangle$ $\displaystyle \left \langle 28,-10,1\right \rangle$

$\left \langle 29,10,3\right \rangle$ $\displaystyle \left \langle 29,10,3\right \rangle$

$\left \langle 15,10,1\right \rangle$ $\displaystyle \left \langle 15,10,1\right \rangle$

$\left \langle 32,40,199\right \rangle$ $\displaystyle \left \langle 32,40,199\right \rangle$

Correct answer:

$\left \langle 28,-10,1\right \rangle$ $\displaystyle \left \langle 28,-10,1\right \rangle$

Explanation:

The formula for the difference of vectors is

$\left \langle x_1,x_2,x_3\right \rangle-\left \langle y_1,y_2,y_3\right \rangle=\left \langle x_1-y_1,x_2-y_2,x_3-y_3\right \rangle$ $\displaystyle \left \langle x_1,x_2,x_3\right \rangle-\left \langle y_1,y_2,y_3\right \rangle=\left \langle x_1-y_1,x_2-y_2,x_3-y_3\right \rangle$

Using the vectors given, we get

$\left \langle 30-2,15-25,100-99\right \rangle=\left \langle 28,-10,1\right \rangle$ $\displaystyle \left \langle 30-2,15-25,100-99\right \rangle=\left \langle 28,-10,1\right \rangle$.

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Example Question #31 : Vector Addition

Find the sum of the vectors $\left \langle 2,0,5\right \rangle$ $\displaystyle \left \langle 2,0,5\right \rangle$ and $\left \langle 1,2,3\right \rangle$ $\displaystyle \left \langle 1,2,3\right \rangle$.

Possible Answers:

$\left \langle 3,2,8\right \rangle$ $\displaystyle \left \langle 3,2,8\right \rangle$

$\left \langle 1,1,5\right \rangle$ $\displaystyle \left \langle 1,1,5\right \rangle$

$\left \langle 1,9,6\right \rangle$ $\displaystyle \left \langle 1,9,6\right \rangle$

$\left \langle 1,7,3\right \rangle$ $\displaystyle \left \langle 1,7,3\right \rangle$

Correct answer:

$\left \langle 3,2,8\right \rangle$ $\displaystyle \left \langle 3,2,8\right \rangle$

Explanation:

The formula for adding vectors is

$a+b=\left \langle a_1+b_1,a_2+b_2,a_3+b_3\right \rangle$ $\displaystyle a+b=\left \langle a_1+b_1,a_2+b_2,a_3+b_3\right \rangle$.

Plugging in the values we were given, we get

$\left \langle (2+1),(0+2),(5+3)\right \rangle=\left \langle 3,2,8\right \rangle$ $\displaystyle \left \langle (2+1),(0+2),(5+3)\right \rangle=\left \langle 3,2,8\right \rangle$

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Example Question #32 : Vector Addition

What is the sum of the vectors $\left \langle 3,9,100\right \rangle$ $\displaystyle \left \langle 3,9,100\right \rangle$ and $\left \langle 2,15,99\right \rangle$ $\displaystyle \left \langle 2,15,99\right \rangle$.

Possible Answers:

224 $\displaystyle 224$

$\left \langle 5,23,100\right \rangle$ $\displaystyle \left \langle 5,23,100\right \rangle$

$\left \langle 2,18,100\right \rangle$ $\displaystyle \left \langle 2,18,100\right \rangle$

$\left \langle 5,24,199\right \rangle$ $\displaystyle \left \langle 5,24,199\right \rangle$

Correct answer:

$\left \langle 5,24,199\right \rangle$ $\displaystyle \left \langle 5,24,199\right \rangle$

Explanation:

The formula for adding vectors is

$a+b=\left \langle a_1+b_1,a_2+b_2,a_3+b_3\right \rangle$ $\displaystyle a+b=\left \langle a_1+b_1,a_2+b_2,a_3+b_3\right \rangle$.

Plugging in the values we were given, we get

$\left \langle (3+2),(9+15),(100+99)\right \rangle=\left \langle 5,24,199\right \rangle$ $\displaystyle \left \langle (3+2),(9+15),(100+99)\right \rangle=\left \langle 5,24,199\right \rangle$

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Example Question #231 : Vectors And Vector Operations

$\vec{a}=6i+4j-2k$ $\displaystyle \vec{a}=6i+4j-2k$ and $\vec{b}=-3i-4j+7k$ $\displaystyle \vec{b}=-3i-4j+7k$. Find $\vec{a}+\vec{b}$ $\displaystyle \vec{a}+\vec{b}$

Possible Answers:

3i+5k $\displaystyle 3i+5k$

$\displaystyle 3i$

$\displaystyle -9i-8j-9k$

$\displaystyle 0$

$\displaystyle 8$

Correct answer:

3i+5k $\displaystyle 3i+5k$

Explanation:

$\vec{a}+\vec{b}=(6-3)i+(4-4)j+(-2+7)k=3i+0j+5k=3i+5k$ $\displaystyle \vec{a}+\vec{b}=(6-3)i+(4-4)j+(-2+7)k=3i+0j+5k=3i+5k$

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Calculus 3 : Vectors and Vector Operations

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #24 : Vector Addition

Example Question #25 : Vector Addition

Example Question #26 : Vector Addition

Example Question #231 : Vectors And Vector Operations

Example Question #28 : Vector Addition

Example Question #29 : Vector Addition

Example Question #30 : Vector Addition

Example Question #31 : Vector Addition

Example Question #32 : Vector Addition

Example Question #231 : Vectors And Vector Operations

All Calculus 3 Resources

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